THE ARBITRAGE PRICING THEORY

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Underlying the arbitrage pricing theory is the acknowledgement that more than one systemic factor would effect the returns on a security an investor may have in his portfolio. This is unlike the capital asset pricing model, which requires only the beta of a security against the market to measure risk.

The arbitrage pricing theory proposes that, the return on a stock would depend on a number of expected and unexpected events which may occur during the time the stock is held by an investor in his portfolio.

Anticipated events would be factored into the price of the stock or security by the investors, and thereby the return they may expect from such stocks and securities. But, the investor would appreciate that, on an average most returns on stocks are realized from unanticipated events which may occur with regard to the stock being held by an investor in his portfolio.

The investor realizes that, change itself is anticipated and that the most unlikely occurrence would be the exact occurrence of a most probable future investment scenario. Although, investors realize that some unexpected events would occur, they do not know their direction or magnitude with regard to price movement of stocks held in their portfolio. The investor would realize and appreciate that systemic factors would have a larger impact on their portfolio performance and returns. Although, the portfolio returns would depend on the same set of systemic factors, but this does not imply identical or similar portfolio performance; as different portfolios (depending on how they have been constructed) would have different levels of sensitivity to these systemic factors.

The investors would appreciate that, as systemic factors are the primary source of risk to their portfolios; it is these very systemic factors that would primarily determine the expected as well as the actual return on the portfolios.

In the arbitrage pricing theory, the actual return R on any security or portfolio may be represented by the following equation:

R = E + bf + e

(Where, R = actual return on a given security or portfolio;
E = expected return on a given security or portfolio;
b = sensitivity of the security to change in the systemic factor;
f = actual return on the systemic factor; and
e = returns on the unsystemic factors).

The above equation would mean that the actual return would equal the expected returns along with factor sensitivity times its factor movement and residual risk. Investors may find it useful to apply three factor or four factor models; which are expected to be appropriate to highlight the influences of the selected systemic factors with regard to stock market returns with respect to their portfolios.

In the instance that an investor decides to apply a four factor model, then the above equation would be suitably expanded to represent the same:

R = E + (b1)(f1) + (b2)(f2) + (b3)(f3) + (b4)(f4) + e

The underlying economic factors thus represented which effect and have influence on the stock market performance would be:

Unanticipated inflation.
Unanticipated changes in expected levels of industrial production.
Unanticipated changes in the risk premiums.
Unanticipated changes in the term structure of interest rates.

Given the above, the main problems faced in the arbitrage pricing theory would pertain to factor identification on the first count; and then separating anticipated and unanticipated movements in these factors with regard to a measurement of their sensitivities.

The investor would find it difficult to determine the exact relationship between the return on their portfolios (and the stocks held in them) and a given risk factor under consideration. Further, it would be even more difficult to determine its sensitivity. In spite of the above, the arbitrage pricing theory does provide intuitive evidence with regard to the way in which stock prices and equilibrium returns are stated; and a similar conclusion may be drawn from the capital asset pricing model.

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